21 For part 1, if there were a subsequence that didn't converge to the same limit, then we could find a neighborhood around the original point such that infinitely-many subsequence terms …

Note that that proposition talks about a convergent sequence, and the behaviour of its subsequences, while $ (b)$ considers any sequence and it's subsequence, and the value it …

The Bolzano-Weierstrass theorem says that any bounded sequence has a subsequence which converges. This does not mean that an unbounded sequence can't have a convergent …

An infinite monochromatic subgraph gives us the indices of a monotone subsequence: If red, the subsequence is increasing while, if blue, it is strictly decreasing.

@user8469759 not necessarily. That would be valid if every Cauchy sequence had a convergent subsequence, but in the OP question we're considering just one sequence.

Every subsequence has a subsequence which converges uniformly. So that isn't immediate.

Is it possible to have a convergent subsequence of a divergent sequence? Ask Question Asked 12 years, 2 months ago Modified 2 years, 1 month ago

Dec 14, 2011 · A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the …

May 21, 2013 · I would like to add that this result easily generalizes to any topological space (as already mentioned). Furthermore, this result implies that in a limit compact space, if every …

Sep 24, 2017 · In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements …